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Mirrors > Home > ILE Home > Th. List > strle2g | GIF version |
Description: Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
Ref | Expression |
---|---|
strle1.i | ⊢ 𝐼 ∈ ℕ |
strle1.a | ⊢ 𝐴 = 𝐼 |
strle2.j | ⊢ 𝐼 < 𝐽 |
strle2.k | ⊢ 𝐽 ∈ ℕ |
strle2.b | ⊢ 𝐵 = 𝐽 |
Ref | Expression |
---|---|
strle2g | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3590 | . 2 ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} = ({〈𝐴, 𝑋〉} ∪ {〈𝐵, 𝑌〉}) | |
2 | strle1.i | . . . . 5 ⊢ 𝐼 ∈ ℕ | |
3 | strle1.a | . . . . 5 ⊢ 𝐴 = 𝐼 | |
4 | 2, 3 | strle1g 12508 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
5 | 4 | adantr 274 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
6 | strle2.k | . . . . 5 ⊢ 𝐽 ∈ ℕ | |
7 | strle2.b | . . . . 5 ⊢ 𝐵 = 𝐽 | |
8 | 6, 7 | strle1g 12508 | . . . 4 ⊢ (𝑌 ∈ 𝑊 → {〈𝐵, 𝑌〉} Struct 〈𝐽, 𝐽〉) |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐵, 𝑌〉} Struct 〈𝐽, 𝐽〉) |
10 | strle2.j | . . . 4 ⊢ 𝐼 < 𝐽 | |
11 | 10 | a1i 9 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → 𝐼 < 𝐽) |
12 | 5, 9, 11 | strleund 12506 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ({〈𝐴, 𝑋〉} ∪ {〈𝐵, 𝑌〉}) Struct 〈𝐼, 𝐽〉) |
13 | 1, 12 | eqbrtrid 4024 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 ∪ cun 3119 {csn 3583 {cpr 3584 〈cop 3586 class class class wbr 3989 < clt 7954 ℕcn 8878 Struct cstr 12412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-struct 12418 |
This theorem is referenced by: strle3g 12510 2strstrg 12518 |
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